I seek advice on whether a certain very long function can be madesignificantly more compact and therefore easier to compute.

I am interested in calculating the value of a function which resemblesboth a continued fraction and an iterated function, where the basicunit of the iteration is the right hand side of the formula forrelativisticaddition of two (collinear) velocities:

(1) W_1 = (u + v) / (1+uv) .

u and v are the summed velocities, given as decimal fractions of thevelocity of light C. Ie: u,v range from 0.0 to 1.0 . Their resultantvelocity is W_n.

The essential idea is to divide C into n <equal> velocities (v) andthenadd them sequentially. My procedure is to take the first sum, W, andthen add to it a third identical velocity v, which gives a secondresultantvelocity

(2) W_2 = (W_1 + v) / (1 + vW_1) .

This is an iteration of the basic form in (1).

Similarly, a fourth identical velocity v is added to W_2, giving

(3) W_3 = (W_2 + v) / (1+ vW_2) ,

and so on and on...

This is to be repeated n times, where n becomes very large andv commensurately smaller. I wish to know the values of this functionas n becomes extremely large. I especially want to know what thefunction converges to as n ---> infinity and C/n ---> zero velocity.

A partial glimpse of this procedure as a single large relationresemblesan unusual variety of continued fraction. I omit rendering it in ASCIIform here since it rapidly becomes very confusing and opaque.